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6-2
Types of estimators
• Point Estimate
A single-valued estimate.
A single element chosen from a sampling distribution.
Conveys little information about the actual value of the
population parameter, about the accuracy of the
estimate.
• Confidence Interval or Interval Estimate
An interval or range of values believed to include the
unknown population parameter.
Associated with the interval is a measure of the
confidence
confidence we have that the interval does indeed
contain the parameter of interest.

6-3
1. Point Estimation
Definition:
 A parameter is a numerical descriptive measure of a
population ( μ is an example of a parameter).
 A statistic is a numerical descriptive measure of a
sample ( X is an example of a statistic).
 To each sample statistic there corresponds a
population parameter.
 We use X , S2
, S , p, etc. to estimate μ, σ2
, σ, P (or π),
etc.

6-4
 Sample statistics
 ˉ (sample mean)
 S2
( sample variance)
 S (sample Standard
deviation)
 P (sample proportion)
 Population
parameter
 μ (population mean)
 σ2
( population
variance)
 σ(population
standard deviation)
 P or π (Population
proportion)
X

6-5
Definition:
 A point estimate of some population parameter O
is a single value Ô of a sample statistic
 Sampling Distribution of Means
 one of the most fundamental concepts of statistical
inference, and it has remarkable properties.
 Since it is a frequency distribution it has its own
mean and standard deviation
 we shall use the notation for the standard
deviation of the distribution.
 The standard deviation of the sampling
distribution of means is called the standard error
of the mean.

6-6
 Properties
1. The mean of the sampling distribution of means is
the same as the population mean, μ .
2. The SD of the sampling distribution of means is σ /
n .
√
3. The shape of the sampling distribution of means is
approximately a normal curve, regardless of the
shape of the population distribution and provided
n is large enough (Central limit theorem).

6-7

Using Statistics

Confidence Interval for the Population Mean When
the Population Standard Deviation is Known

Confidence Intervals for  When  is Unknown - The
t Distribution

Large-Sample Confidence Intervals for the
Population Proportion p

Sample Size Determination
Confidence Intervals

6-8
• Consider the following statements:
x = 550
• A single-valued estimate that conveys little information
about the actual value of the population mean.
We are 99% confident that  is in the interval [449,551]
• An interval estimate which locates the population mean
within a narrow interval, with a high level of confidence.
We are 90% confident that  is in the interval [400,700]
• An interval estimate which locates the population mean
within a broader interval, with a lower level of confidence.

6-9
A confidence interval or interval estimate is a range or interval of
numbers believed to include an unknown population parameter.
Associated with the interval is a measure of the confidence we have
that the interval does indeed contain the parameter of interest.
• A confidence interval or interval estimate
has two components:
A range or interval of values
An associated level of confidence

6-10

If the population distribution is normal, the
sampling distribution of the mean is normal.
 If the sample is sufficiently large, regardless of the
shape of the population distribution, the sampling
distribution is normal (Central Limit Theorem).
95
.
0
96
.
1
96
.
1
or
95
.
0
96
.
1
96
.
1
:
case
either
In






















n
x
n
x
P
n
x
n
P







4
3
2
1
0
-1
-2
-3
-4
0.4
0.3
0.2
0.1
0.0
z
f(z)
Standard Normal Distribution: 95% Interval

6-11
.
is,
That
not).
will
them
of
5%
(and
mean
population
the
include
will
96
.
1
intervals
such
of
95%
ely
approximat
sampling,
After



for
interval
confidence
95%
a
is
n
1.96
x 

n
x

6-12
Approximately 95% of the intervals
around the sample mean can be
expected to include the actual value of the
population mean, . (When the sample
mean falls within the 95% interval around
the population mean.)
*5% of such intervals around the sample
mean can be expected not
not to include the
actual value of the population mean.
(When the sample mean falls outside the
95% interval around the population
mean.)
x x
x
n
x 
96
.
1

0.4
0.3
0.2
0.1
0.0
x
f(x)
Sampling Distribution of the Mean

x
x
x
x
x
x
x
x
2.5%
95%
2.5%


 196
.
n


196
.
n
x
x
x
*
*
p n u
95%

6-13
Interpretation:
a.Probabilistic: in repeated sampling, 100(1-α)% of all
intervals will include μ
b.Practical: we are 100(1-α)% confident that a interval
contains μ.

6-14
A 95% confidence interval for  when  is known and sampling is
done from a normal population, or a large sample is used:
n
x

96
.
1

The quantity is often called the margin of error or the
sampling error.
n

96
.
1
For example, if:n = 25
= 20
= 122
 
84
.
129
,
16
.
114
84
.
7
122
)
4
)(
96
.
1
(
122
25
20
96
.
1
122
96
.
1








n
x

A 95% confidence interval:
x

6-15
2

z
2

z
2
( )
1 
2

z


2

2

6-16
0.99 0.005 2.576
0.98 0.010 2.326
0.95 0.025 1.960
0.90 0.050 1.645
0.80 0.100 1.282
( )
1 

2
z
2
z
2
( )
1  
 z
2

2

2

6-17
When sampling from the same population, using a fixed sample size, the
higher the confidence level, the wider the confidence interval.
5
4
3
2
1
0
-1
-2
-3
-4
-5
0.4
0.3
0.2
0.1
0.0
Z
f(z)
Stand ard Nor m al Distribution
80% Confidence Interval:
x
n
128
.

5
4
3
2
1
0
-1
-2
-3
-4
-5
0.4
0.3
0.2
0.1
0.0
Z
f(z)
Stand ard Nor m al Distributi on
95% Confidence Interval:
x
n
196
.


6-18
When sampling from the same population, using a fixed confidence
level, the larger the sample size, n, the narrower the confidence
interval.
0 .9
0 .8
0 .7
0 .6
0 .5
0 .4
0 .3
0 .2
0 .1
0 .0
x
f(x)
S am p ling D istrib utio n of the M e an
95% Confidence Interval: n = 40
0.4
0.3
0.2
0.1
0.0
x
f(x)
S am p ling D istrib utio n of the Me an
95% Confidence Interval: n = 20
0 .9
0 .8
0 .7
0 .6
0 .5
0 .4
0 .3
0 .2
0 .1
0 .0
x
f(x)
0 .4
0 .3
0 .2
0 .1
0 .0
x
f(x)
0 .9
0 .8
0 .7
0 .6
0 .5
0 .4
0 .3
0 .2
0 .1
0 .0
x
f(x)
0 .4
0 .3
0 .2
0 .1
0 .0
x
f(x)

6-19
 A physical therapist wished to estimate, with
99% confidence, the mean maximal strength
of a particular muscle in a certain group of
individuals. He assume that strength scores
are approximately normally distributed with
a variance of 144. A sample of 15 subjects
who participated in the experiment yielded a
mean of 84.3. What is 90% CI?

6-20
Solution
α = 0.01⇒ Zα/2 = 2.58
Mean =84.3, n=15, σ =12
84.3 ± 2.58(12/ √15) 84.3 ± 8.0 (76.3, 92.3)
⇒ ⇒
⇒ We are 99% confident that the population mean is
between 76.3 and 92.3.

6-21
• The t is a family of bell-shaped and symmetric
distributions, one for each number of degree of
freedom.
• The expected value of t is 0.
• For df > 2, the variance of t is df/(df-2). This is
greater than 1, but approaches 1 as the number
of degrees of freedom increases. The t is flatter
and has fatter tails than does the standard
normal.
• The t distribution approaches a standard normal
as the number of degrees of freedom increases
If the population standard deviation, , is not known, replace
with the sample standard deviation, s. If the population is
normal, the resulting statistic:
has a t distribution with (n - 1) degrees of freedom.
t
X
s
n

 
Standard normal
t, df = 20
t, df = 10



6-22
A (1-)100% confidence interval for  when  is not known
(assuming a normally distributed population):
where is the value of the t distribution with n-1 degrees of
freedom that cuts off a tail area of to its right.
t
2 
2
n
s
t
x
2



6-23
df t0.100 t0.050 t0.025 t0.010 t0.005
--- ----- ----- ------ ------ ------
1 3.078 6.314 12.706 31.821 63.657
2 1.886 2.920 4.303 6.965 9.925
3 1.638 2.353 3.182 4.541 5.841
4 1.533 2.132 2.776 3.747 4.604
5 1.476 2.015 2.571 3.365 4.032
6 1.440 1.943 2.447 3.143 3.707
7 1.415 1.895 2.365 2.998 3.499
8 1.397 1.860 2.306 2.896 3.355
9 1.383 1.833 2.262 2.821 3.250
10 1.372 1.812 2.228 2.764 3.169
11 1.363 1.796 2.201 2.718 3.106
12 1.356 1.782 2.179 2.681 3.055
13 1.350 1.771 2.160 2.650 3.012
14 1.345 1.761 2.145 2.624 2.977
15 1.341 1.753 2.131 2.602 2.947
16 1.337 1.746 2.120 2.583 2.921
17 1.333 1.740 2.110 2.567 2.898
18 1.330 1.734 2.101 2.552 2.878
19 1.328 1.729 2.093 2.539 2.861
20 1.325 1.725 2.086 2.528 2.845
21 1.323 1.721 2.080 2.518 2.831
22 1.321 1.717 2.074 2.508 2.819
23 1.319 1.714 2.069 2.500 2.807
24 1.318 1.711 2.064 2.492 2.797
25 1.316 1.708 2.060 2.485 2.787
26 1.315 1.706 2.056 2.479 2.779
27 1.314 1.703 2.052 2.473 2.771
28 1.313 1.701 2.048 2.467 2.763
29 1.311 1.699 2.045 2.462 2.756
30 1.310 1.697 2.042 2.457 2.750
40 1.303 1.684 2.021 2.423 2.704
60 1.296 1.671 2.000 2.390 2.660
120 1.289 1.658 1.980 2.358 2.617
1.282 1.645 1.960 2.326 2.576

0
0 .4
0 .3
0 .2
0 .1
0 .0
t
f(t)
t D istrib utio n: d f=10
Area = 0.10
}
Area = 0.10
}
Area = 0.025
}
Area = 0.025
}
1.372
-1.372
2.228
-2.228
Whenever  is not known (and the population is
assumed normal), the correct distribution to use is
the t distribution with n-1 degrees of freedom.
Note, however, that for large degrees of freedom,
the t distribution is approximated well by the Z
distribution.

6-24
A study of hypoxemia during the immediate post-operative period reported
the fractions of ideal weight for 11 patients who became severely hypoxemic
during transfer to the recovery room. The mean is 1.51 and the standard
deviation is 0.33. Estimate the 95% C.I. for the population mean fraction of
ideal weight, where the population consists of hypoxemic patients similar to
those in the study (The data is normally distributed, use α=0.05).
Solution
• tα/2, n-1 / = t 0.025,10 = 2.2281
1 . 51 ± 2 . 2281(0 . 33/11)
1 . 51± 0 . 221
(1 . 289 ,1 . 731 )
• We are 95% sure that the μ (1 . 289 ,1 . 731 ) population mean lies
between 1.289 and 1.731

6-25
Whenever  is not known (and the population is
assumed normal), the correct distribution to use is
the t distribution with n-1 degrees of freedom.
Note, however, that for large degrees of freedom,
the t distribution is approximated well by the Z
distribution.
df t0.100 t0.050 t0.025 t0.010 t0.005
--- ----- ----- ------ ------ ------
1 3.078 6.314 12.706 31.821 63.657
. . . . . .
. . . . . .
. . . . . .
120 1.289 1.658 1.980 2.358 2.617
1.282 1.645 1.960 2.326 2.576


6-26
n
s
z
x
2
:
for
interval
confidence
)100%
-
(1
sample
-
large
A




Example 6-3:
Example 6-3: An economist wants to estimate the average amount in checking accounts
at banks in a given region. A random sample of 100 accounts gives x-bar = $357.60
and s = $140.00. Give a 95% confidence interval for , the average amount in any
checking account at a bank in the given region.
 
x z
s
n
     
0 025
357.60 196
14000
100
357.60 27.44 33016,38504
.
.
.
. .

6-27
The estimator of the population proportion, , is the sample proportion, . If the
sample size is large,
p p
p p p
p
pq
n
q = (1 - p)
p p
p n p n q

 

 
has an approximately normal distribution, with E( ) = and
V( ) = where . When the population proportion is unknown, use the
estimated value, , to estimate the standard deviation of .
For estimating , a sample is considered large enough when both an are greater
than 5.
,
 

6-28
.
p̂
-
1
=
q̂
and
,
size),
sample
(the
trials
of
number
by the
divided
,
sample,
in the
successes
of
number
the
to
equal
is
,
p̂
,
proportion
sample
the
where
,
proportion
population
for the
interval
confidence
)100%
-
(1
sample
-
large
A
ˆ
ˆ
2
ˆ
n
x
:
p
n
q
p
α
z
p 


6-29
A marketing research firm wants to estimate the share that foreign companies
have in the American market for certain products. A random sample of 100
consumers is obtained, and it is found that 34 people in the sample are users
of foreign-made products; the rest are users of domestic products. Give a
95% confidence interval for the share of foreign products in this market.
 

 
. .
( . )( . )
. ( . )( . )
. .
. , .
p z
pq
n
  
 
 


2
0 34 196
0 34 0 66
100
0 34 196 0 04737
0 34 0 0928
0 2472 0 4328
Thus, the firm may be 95% confident that foreign manufacturers control
anywhere from 24.72% to 43.28% of the market.

6-30
The width of a confidence interval can be reduced only at the
price of:
• a lower level of confidence, or
• a larger sample.
 

 
. .
( . )( . )
. ( . )( . )
. .
. , .
p z
pq
n
  
 
 


2
0 34 1645
0 34 0 66
100
0 34 1645 0 04737
0 34 0 07792
0 2621 0 4197
90% Confidence Interval
 

 
. .
( . )( . )
. ( . )( . )
. .
. , .
p z
pq
n
  
 
 


2
0 34 196
0 34 0 66
200
0 34 196 0 03350
0 34 0 0657
0 2743 0 4057
Sample Size, n = 200
Lower Level of Confidence Larger Sample Size

6-31
• How close do you want your sample estimate to be to the
unknown parameter? (What is the desired bound, B?)
• What do you want the desired confidence level (1-) to be so
that the distance between your estimate and the parameter is
less than or equal to B?
• What is your estimate of the variance (or standard deviation)
of the population in question?
Before determining the necessary sample size, three questions must
be answered:
n


 
2
z
x
:
for
Interval
Confidence
)
-
(1
A
:
example
For 

6-32

Standard error
of statistic
Sample size = n
Sample size = 2n
Standard error
of statistic
The sample size determines the bound of a statistic, since the standard
error of a statistic shrinks as the sample size increases:

6-34
A marketing research firm wants to conduct a survey to estimate the average
amount spent on entertainment by each person visiting a popular resort. The
people who plan the survey would like to determine the average amount spent by
all people visiting the resort to within $120, with 95% confidence. From past
operation of the resort, an estimate of the population standard deviation is
s = $400. What is the minimum required sample size?
n
z
B


 


2
2 2
2
2 2
2
1 96 400
120
42 684 43
( . ) ( )
.

6-35
The manufacturers of a sports car want to estimate the proportion of people in a
given income bracket who are interested in the model. The company wants to
know the population proportion, p, to within 0.01 with 99% confidence. Current
company records indicate that the proportion p may be around 0.25. What is the
minimum required sample size for this survey?
n
z pq
B


 

2
2
2
2
2
2 576 025 0 75
010
124.42 125
. ( . )( . )
.

�ݺ�ߣ

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